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We construct an approximate renormalization for Hamiltonian systems with twodegrees of freedom in order to study the break-up of invariant tori witharbitrary frequency. We derive the equation of the critical surface of therenormalization map, and we compute the scaling behavior of the criticalfunction of one-parameter families of Hamiltonians, near rational frequencies.For the forced pendulum model, we find the same scaling law found for thestandard map in [Carletti and Laskar, preprint (2000)]. We discuss a conjectureon the link between the critical function of various types of forced pendulummodels, with the Bruno function.Comment: latex article.tex, 5 files, 14 pages, submitted to Nonlinearit

Scaling law for the critical function of an approximate renormalization